Went to Coombe Wood (borough of Croydon) a fortnight ago with the kids. Nothing child specific but it is the right size for a walk with younger ones. There are gardens with wel kept beds, rockeries and a wooded area. We made a den. The cafe has a children’s menu and my food was much above average for a park cafe. Lots of ice cream flavours too.
We also had a round of mini golf at “dragon golf” recently which is nearish Coombe wood. Bit run down and dragon doesn’t work, but hey the kids loved it and during half term it was mega quiet.